Методические указания по выполнению контрольной работы № 2 по математике для студентов инженерно-технических специальностей заочной формы обучения
.pdfy* |
x cos x x2 sin x . |
|
|
|
|
|
|
||
Ɉɛɳɟɟ |
ɪɟɲɟɧɢɟ |
ɛɭɞɟɬ y |
y y C cos x C |
2 |
sin x x cos x x2 sin x . |
||||
|
|
|
|
|
|
1 |
|
|
|
ɇɚɯɨɞɢɦ |
yc |
C1 sin x C2 cos x cos x x sin x 2x sin x x2 cos x . |
Ɍɚɤ ɤɚɤ |
||||||
|
c |
1, ɬɨ 0 |
C1, C |
C2 1. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, C1 |
|
0, C2 0 . ɉɨɞɫɬɚɜɥɹɹ |
|||
y(0) 0, y (0) |
|
||||||||
ɡɧɚɱɟɧɢɹ |
C1 |
0, C2 |
0 |
ɜ |
ɨɛɳɟɟ |
ɪɟɲɟɧɢɟ, ɩɨɥɭɱɢɦ ɱɚɫɬɧɨɟ |
ɪɟɲɟɧɢɟ |
||
y x cos x x2 sin x .
ɉɪɢɦɟɪ 6.4. Ɉɩɪɟɞɟɥɢɬɶ ɜɢɞ ɱɚɫɬɧɨɝɨ ɪɟɲɟɧɢɹ ɥɢɧɟɣɧɨɝɨ ɧɟɨɞɧɨɪɨɞɧɨɝɨ
ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ, ɟɫɥɢ ɢɡɜɟɫɬɧɵ ɤɨɪɧɢ k1 |
3 2i , |
k2 |
3 2i ɟɝɨ |
||
ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ ɢ ɟɝɨ ɩɪɚɜɚɹ ɱɚɫɬɶ |
|
|
|
||
f (x) |
e3x (cos 2x sin 2x) . |
|
|
|
|
Ɋɟɲɟɧɢɟ. ȼ ɩɪɚɜɨɣ ɱɚɫɬɢ D |
3, E 2, Pn ( x) 1, Qm ( x) 1 |
– |
ɦɧɨɝɨɱɥɟɧɵ |
||
ɧɭɥɟɜɨɣ |
ɫɬɟɩɟɧɢ, DrEi 3 r2i |
ɹɜɥɹɸɬɫɹ ɤɨɪɧɹɦɢ |
ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɝɨ |
||
ɭɪɚɜɧɟɧɢɹ. ɉɨɷɬɨɦɭ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ |
|
|
|
||
y |
xe3x ( Acos 2x Bsin 2x) , |
|
|
|
|
ɝɞɟ A ɢ B – ɧɟɨɩɪɟɞɟɥɟɧɧɵɟ ɤɨɷɮɮɢɰɢɟɧɬɵ.
7. ɋɂɋɌȿɆɕ ȾɂɎɎȿɊȿɇɐɂȺɅɖɇɕɏ ɍɊȺȼɇȿɇɂɃ. ɆȿɌɈȾ ɂɋɄɅɘɑȿɇɂə. ɆȿɌɈȾ ɗɃɅȿɊȺ Ɋȿɒȿɇɂə ɅɂɇȿɃɇɕɏ ɋɂɋɌȿɆ ɋ ɉɈɋɌɈəɇɇɕɆɂ ɄɈɗɎɎɂɐɂȿɇɌȺɆɂ
7.1 ɇɨɪɦɚɥɶɧɚɹ ɫɢɫɬɟɦɚ n–ɝɨ ɩɨɪɹɞɤɚ ɨɛɵɤɧɨɜɟɧɧɵɯ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ
ɇɨɪɦɚɥɶɧɚɹ ɫɢɫɬɟɦɚ n–ɝɨ ɩɨɪɹɞɤɚ ɨɛɵɤɧɨɜɟɧɧɵɯ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɢɦɟɟɬ ɜɢɞ
71
|
dx1 |
|
|
f1(t, x1, x2 ,..., xn ); |
|
|
|||
|
° |
|
|
|
|
|
|
|
|
|
|
dt |
|
|
|
|
|||
|
|
|
|
|
|
|
|||
|
°dx |
2 |
|
|
|
|
|
||
|
° |
|
|
|
|
f2 (t, x1, x2 ,..., xn ); |
|
|
|
|
|
dt |
|
|
|
|
|||
|
® |
|
|
|
... |
|
|
||
|
° |
|
|
|
|
|
|
|
|
|
°dx |
n |
|
|
fn (t, x1, x2 ,..., xn ). |
|
|
||
|
° |
|
|
|
|
|
|
||
|
|
|
|
|
|
|
|
||
|
¯ dt |
|
|
|
|
|
|||
ɝɞɟ |
t |
– |
|
ɧɟɡɚɜɢɫɢɦɚɹ ɩɟɪɟɦɟɧɧɚɹ; x1, x2 ,..., xn |
– |
ɧɟɢɡɜɟɫɬɧɵɟ ɮɭɧɤɰɢɢ ɨɬ |
|||
t; f1, |
f2 ,..., |
|
fn |
– ɡɚɞɚɧɧɵɟ ɮɭɧɤɰɢɢ. |
|
|
|||
|
Ɇɟɬɨɞ |
ɢɫɤɥɸɱɟɧɢɹ ɧɟɢɡɜɟɫɬɧɵɯ ɫɨɫɬɨɢɬ |
ɜ |
ɬɨɦ, ɱɬɨ ɞɚɧɧɚɹ ɫɢɫɬɟɦɚ |
|||||
ɩɪɢɜɨɞɢɬɫɹ ɤ ɨɞɧɨɦɭ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɦɭ ɭɪɚɜɧɟɧɢɸ n–ɝɨ ɩɨɪɹɞɤɚ ɫ ɨɞɧɨɣ ɧɟɢɡɜɟɫɬɧɨɣ ɮɭɧɤɰɢɟɣ (ɢɥɢ ɤ ɧɟɫɤɨɥɶɤɢɦ ɭɪɚɜɧɟɧɢɹɦ, ɫɭɦɦɚ ɩɨɪɹɞɤɨɜ ɤɨɬɨɪɵɯ ɪɚɜɧɚ n). Ⱦɥɹ ɷɬɨɝɨ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɞɢɮɮɟɪɟɧɰɢɪɭɸɬ ɨɞɧɨ ɢɡ ɭɪɚɜɧɟɧɢɣ ɫɢɫɬɟɦɵ ɢ ɢɫɤɥɸɱɚɸɬ ɜɫɟ ɧɟɢɡɜɟɫɬɧɵɟ ɮɭɧɤɰɢɢ, ɤɪɨɦɟ ɨɞɧɨɣ.
ɉɪɢɦɟɪ 7.1. ɇɚɣɬɢ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɫɢɫɬɟɦɵ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ
dx y |
, |
dy y(x 2 y 1) |
||||
|
|
|
|
|
|
|
dt t |
dt |
|
t(x 1) |
|||
|
|
|||||
ɢ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ, ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɟ ɧɚɱɚɥɶɧɵɦ ɭɫɥɨɜɢɹɦ x(1)
Ɋɟɲɟɧɢɟ. Ⱦɢɮɮɟɪɟɧɰɢɪɭɟɦ ɩɟɪɜɨɟ ɭɪɚɜɧɟɧɢɟ ɩɨ t: xcc
1; y(1) 4 .
yct y . Ɂɚɦɟɧɹɹ t2
ɡɞɟɫɶ yc ɟɟ ɡɧɚɱɟɧɢɟɦ ɢɡ ɜɬɨɪɨɝɨ ɭɪɚɜɧɟɧɢɹ ɫɢɫɬɟɦɵ ɢ ɩɨɞɫɬɚɜɥɹɹ y xct ,
ɧɚɣɞɟɧɧɨɟ ɢɡ ɩɟɪɜɨɝɨ ɭɪɚɜɧɟɧɢɹ, ɩɨɥɭɱɢɦ ɩɨɫɥɟ ɭɩɪɨɳɟɧɢɹ ɭɪɚɜɧɟɧɢɟ ɜɬɨɪɨɝɨ
2(xc)2 . x 1
ɂɧɬɟɝɪɢɪɭɟɦ ɷɬɨ ɭɪɚɜɧɟɧɢɟ, ɩɪɟɞɜɚɪɢɬɟɥɶɧɨ ɩɨɧɢɠɚɹ ɩɨɪɹɞɨɤ:
xc p; |
p p(x); xcc |
|
dp |
|
dp |
|
|
2 p |
|
|
dp |
2dx |
|
|
|
||||||
|
|
p; |
|
|
|
|
|
|
; |
|
|
|
; |
|
|
||||||
|
dx |
|
dx |
|
|
x 1 |
|
dx |
|
x 1 |
C1t C2 1 |
|
|||||||||
p C |
(x 1)2 ; |
dx |
C |
(x 1)2 |
; |
1 |
|
|
C t C |
2 |
; x |
. |
|||||||||
|
|
|
|
||||||||||||||||||
1 |
|
dt |
1 |
|
|
|
|
|
|
x 1 |
|
1 |
|
|
|
C1t C2 |
|||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||
72
Ⱦɢɮɮɟɪɟɧɰɢɪɭɹ ɷɬɭ ɮɭɧɤɰɢɸ ɢ ɩɨɞɫɬɚɜɥɹɹ ɜ ɜɵɪɚɠɟɧɢɟ y xct , ɩɨɥɭɱɢɦ
|
y |
|
|
|
C1t |
|
|
|
. |
|
|
|
|
|
|
|
|
|
|
|
|
||
|
|
(C t C |
2 |
)2 |
|
|
|
|
|
|
|
|
|
|
|
|
|||||||
|
|
|
|
|
1 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Ɉɛɳɢɦ ɪɟɲɟɧɢɟɦ ɞɚɧɧɨɣ ɫɢɫɬɟɦɵ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɛɭɞɟɬ |
||||||||||||||||||||||
|
x |
|
C1t C2 1 |
, |
y |
C1t |
|
|
. |
|
|
|
|
|
|
|
|||||||
|
|
|
|
C1t C2 |
(C t C |
2 |
)2 |
|
|
|
|
|
|
|
|||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
1 |
|
|
|
|
|
|
|
|
|
|
|
Ⱦɥɹ ɧɚɯɨɠɞɟɧɢɹ ɱɚɫɬɧɨɝɨ ɪɟɲɟɧɢɹ ɩɨɞɫɬɚɜɢɦ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ |
||||||||||||||||||||||
|
x(1) |
|
1, |
|
|
y(1) |
|
4 . |
ɉɨɥɭɱɢɦ 1 |
C1 C2 |
1 |
; 4 |
C1 |
|
|
, ɨɬɤɭɞɚ |
|||||||
|
|
|
|
|
C C |
2 |
(C C |
2 |
) |
||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1 |
|
1 |
|
|
||
C |
1, |
C |
2 |
|
1 |
. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||||||
1 |
|
|
2 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɢɫɤɨɦɵɦ ɱɚɫɬɧɵɦ ɪɟɲɟɧɢɟɦ ɫɢɫɬɟɦɵ ɛɭɞɟɬ ɩɚɪɚ ɮɭɧɤɰɢɣ:
|
x |
2t 3 |
, |
y |
|
4t |
. |
|
|
|||
|
(2t 12 |
|
|
|||||||||
|
|
|
2t 1 |
|
|
|
|
|||||
ɉɪɢɦɟɪ 7.2. ɇɚɣɬɢ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɫɢɫɬɟɦɵ |
|
|
||||||||||
|
dx |
2 y 5x et , |
|
dy |
x 6 y e 2t . |
|
|
|||||
|
|
|
|
|
|
|
|
|||||
|
dt |
|
dt |
|
|
|||||||
|
|
|
|
|
|
|
|
|
||||
Ɋɟɲɟɧɢɟ. |
Ⱦɢɮɮɟɪɟɧɰɢɪɭɟɦ ɩɟɪɜɨɟ ɭɪɚɜɧɟɧɢɟ: xcc 2 yc 5xc et . Ɂɚɦɟɧɹɟɦ |
|||||||||||
yc ɟɟ ɡɧɚɱɟɧɢɟɦ ɢɡ ɜɬɨɪɨɝɨ ɭɪɚɜɧɟɧɢɹ ɢ ɩɨɞɫɬɚɜɥɹɟɦ ɡɚɬɟɦ y |
1 |
(xc 5x et ) . |
||||||||||
2 |
||||||||||||
ɉɨɥɭɱɢɦ ɥɢɧɟɣɧɨɟ ɧɟɨɞɧɨɪɨɞɧɨɟ ɭɪɚɜɧɟɧɢɟ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɫ ɩɨɫɬɨɹɧɧɵɦɢ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ
xcc 11xc 28x 2e 2t 7et . |
|
|
|
|||||||||
ȿɝɨ ɨɛɳɟɟ ɪɟɲɟɧɢɟ |
|
|
|
|
|
|
|
|
|
|||
x C e 4t C |
2 |
e 7t |
|
1 |
e 2t |
7 |
et |
|
|
|
||
|
|
|
|
|
||||||||
1 |
|
|
2 |
|
40 |
|
|
|
|
|||
|
|
|
|
|
|
|
|
|
||||
(ɩɨɥɭɱɟɧɨ ɤɚɤ |
ɫɭɦɦɚ ɨɛɳɟɝɨ |
ɪɟɲɟɧɢɹ x C e 4t C |
2 |
e 7t |
ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɝɨ |
|||||||
|
|
|
|
|
|
|
1 |
|
|
|||
73
ɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ |
ɢ ɱɚɫɬɧɨɝɨ ɪɟɲɟɧɢɹ x* |
|
1 |
e |
2t |
|
7 |
e |
t |
ɧɟɨɞɧɨɪɨɞɧɨɝɨ |
|||||||||||||||||||||||
5 |
|
|
|
40 |
|
||||||||||||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||
ɭɪɚɜɧɟɧɢɹ). |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
ɉɨɞɫɬɚɜɥɹɹ x ɢ xc |
ɜ ɜɵɪɚɠɟɧɢɟ ɞɥɹ y, ɩɨɥɭɱɢɦ |
|
|
|
|
|
|
|
|
|
|
|
|
||||||||||||||||||||
y |
1 |
(xc 5x |
e |
t |
) |
1 |
C1e |
4t |
C2 e |
7t |
|
3 |
e |
2t |
|
|
1 |
|
e |
t |
. |
|
|
|
|
||||||||
2 |
|
|
|
2 |
|
|
|
|
10 |
|
40 |
|
|
|
|
|
|||||||||||||||||
Ɉɛɳɟɟ ɪɟɲɟɧɢɟ ɢɫɯɨɞɧɨɣ ɫɢɫɬɟɦɵ ɢɦɟɟɬ ɜɢɞ |
|
|
|
|
|
|
|
|
|
|
|
|
|||||||||||||||||||||
x C e 4t C |
2 |
e 7t |
|
1 |
e 2t |
7 |
et ; |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||||||||||
|
1 |
|
|
|
5 |
|
|
40 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||
y12 C1e 4t C2e 7t 103 e 2t 401 et .
7.2.Ʌɢɧɟɣɧɚɹ ɨɞɧɨɪɨɞɧɚɹ ɫɢɫɬɟɦɚ n–ɝɨ ɩɨɪɹɞɤɚ
ɫɩɨɫɬɨɹɧɧɵɦɢ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ
Ʌɢɧɟɣɧɚɹ ɨɞɧɨɪɨɞɧɚɹ ɫɢɫɬɟɦɚ n–ɝɨ ɩɨɪɹɞɤɚ ɫ ɩɨɫɬɨɹɧɧɵɦɢ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ ɢɦɟɟɬ ɜɢɞ
dx1 ° dt
°°dx2 ®° dt
°°dxn
¯ dt
a11x1 |
a12 x2 ... a1n xn ; |
|
a21x1 |
a22 x2 |
... a2n xn ; |
|
... |
|
an1x1 |
an2 x2 |
... ann xn , |
ɝɞɟ aij const, aij R, xi – ɧɟɢɡɜɟɫɬɧɵɟ ɮɭɧɤɰɢɢ ɨɬ t.
Ⱦɚɧɧɭɸ ɫɢɫɬɟɦɭ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɦɚɬɪɢɱɧɨɣ ɮɨɪɦɟ
|
dX |
AX , |
|
dt |
|
|
|
|
ɝɞɟ |
|
|
74
|
§a |
a |
a |
· |
|
§ x |
· |
|
|
||
|
¨ |
11 |
12 |
1n ¸ |
|
¨ |
1 |
¸ |
|
|
|
A |
¨a21 |
a22 |
a2n ¸ |
; X |
¨x2 |
¸ |
; |
dX |
|||
¨ |
|
|
|
¸ |
¨ |
|
¸ |
dt |
|||
|
¨ |
|
|
|
¸ |
|
¨ |
|
¸ |
|
|
|
¨ |
|
an2 |
|
¸ |
|
¨ |
|
¸ |
|
|
|
©an1 |
ann ¹ |
|
©xn |
¹ |
|
|
||||
§¨ dx1 ·¸ ¨ dt ¸ ¨ dx2 ¸ ¨¨ dt ¸¸. ¨¨©dxdtn ¸¸¹
ɉɪɢ ɪɟɲɟɧɢɢ ɥɢɧɟɣɧɨɣ ɫɢɫɬɟɦɵ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɦɟɬɨɞɨɦ
ɗɣɥɟɪɚ ɱɚɫɬɧɵɟ ɪɟɲɟɧɢɹ ɫɢɫɬɟɦɵ ɢɳɭɬɫɹ ɜ ɜɢɞɟ |
X |
Vekt , ɝɞɟ V z 0 – ɦɚɬɪɢɰɚ– |
||||
ɫɬɨɥɛɟɰ, k j – ɱɢɫɥɨ. |
|
|
|
|
||
ȿɫɥɢ ɤɨɪɧɢ k1, k2 ,..., kn |
ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɝɨ |
ɭɪɚɜɧɟɧɢɹ |
det(A kE) 0 |
|||
ɞɟɣɫɬɜɢɬɟɥɶɧɵ ɢ ɪɚɡɥɢɱɧɵ, ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɫɢɫɬɟɦɵ ɢɦɟɟɬ ɜɢɞ |
|
|||||
X C V ek1t C V ek2t ... C V eknt , |
|
|
|
|||
1 |
1 |
2 2 |
n n |
|
|
|
C1,C2 ,...,Cn |
– |
ɩɪɨɢɡɜɨɥɶɧɵɟ |
ɩɨɫɬɨɹɧɧɵɟ, V j – |
ɫɨɛɫɬɜɟɧɧɵɣ |
ɜɟɤɬɨɪ–ɫɬɨɥɛɟɰ |
|
ɦɚɬɪɢɰɵ A, ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɣ ɱɢɫɥɭ k, ɬɨ ɟɫɬɶ ( A k j E)V j 0 , ɝɞɟ E – ɟɞɢɧɢɱɧɚɹ ɦɚɬɪɢɰɚ.
Ɂɚɦɟɱɚɧɢɟ. ȿɫɥɢ km ,km – ɩɚɪɚ ɩɪɨɫɬɵɯ ɤɨɦɩɥɟɤɫɧɨ–ɫɨɩɪɹɠɟɧɧɵɯ ɤɨɪɧɟɣ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ, ɬɨ ɢɦ ɫɨɨɬɜɟɬɫɬɜɭɸɬ ɞɜɚ ɞɟɣɫɬɜɢɬɟɥɶɧɵɯ
ɱɚɫɬɧɵɯ ɪɟɲɟɧɢɹ Re(Vmekmt ); Im(Vmekmt ) , ɝɞɟ Re z, |
Im z |
– ɞɟɣɫɬɜɢɬɟɥɶɧɵɟ ɢ |
||
ɦɧɢɦɵɟ ɱɚɫɬɢ z. |
|
|
||
ɉɪɢɦɟɪ 7.3. ɇɚɣɬɢ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɫɢɫɬɟɦɵ |
|
|
||
dx |
x 2 y 2z; |
|
|
|
° |
|
|
|
|
|
|
|
|
|
°dt |
|
|
|
|
°dy |
x 4 y 2z; |
|
|
|
® |
|
|
|
|
°dt |
|
|
|
|
° |
dz |
x 5y 3z, |
|
|
° |
|
|
|
|
¯dt |
|
|
|
|
ɢ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ, ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɟ ɭɫɥɨɜɢɹɦ x(0) |
1, y(0) |
2 , z(0) 0 . |
||
Ɋɟɲɟɧɢɟ. ɋɨɫɬɚɜɥɹɟɦ ɢ ɪɟɲɚɟɦ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ
75
|
1 k |
|
2 |
|
|
|
2 |
|
|
|
|
|
|
(k 2 k 2)(1 k) |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||||||||||||||
|
1 |
|
|
4 k |
|
|
2 |
|
|
0, |
|
|
|
0, k |
|
|
1, |
|
k |
2 |
1, |
|
k |
3 |
2. |
|||||||||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1 |
|
|
|
|
|
|
|
|
|
|
|
|
|||
|
1 |
|
|
|
|
5 |
|
|
|
3k |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||
ɇɚɯɨɞɢɦ ɫɨɛɫɬɜɟɧɧɵɣ ɜɟɤɬɨɪ V1 , ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɣ ɤɨɪɧɸ k1 |
|
1: |
|
|||||||||||||||||||||||||||||||||||||||||||||||
|
|
|
§v |
· |
|
§1 |
|
|
( 1) |
|
|
|
2 |
2 |
·§v |
· §0 |
· |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||||||||||||
V1 |
|
¨ |
1 |
¸ |
|
¨ |
|
|
|
|
|
|
4 |
|
|
|
( 1) 2 |
¸¨ |
1 |
¸ |
¨ |
0 |
¸ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||||||
|
¨v2 |
¸; |
¨1 |
|
|
|
|
|
|
|
|
|
¸¨v2 |
¸ ¨ |
¸ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||||||||||||||||
|
|
|
¨v |
¸ |
|
¨1 |
|
|
|
|
|
|
5 |
|
|
|
|
|
3 |
( 1) |
¸¨v |
¸ ¨ |
0 |
¸ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||||||
|
|
|
© |
3 |
¹ |
|
© |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
¹© |
3 |
¹ |
© |
|
|
¹ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||
2v |
2v |
2 |
|
2v |
|
|
0; |
|
|
|
v |
2 |
v ; |
|
|
|
§ 1 |
|
· |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||||||||||||
° |
1 |
|
|
|
|
|
|
|
3 |
|
0; |
° |
|
1 |
|
|
|
¨ |
|
|
|
1 |
¸ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||||||||
® v1 5v2 2v3 |
|
|
®v3 |
2v1; V1 |
¨ |
|
¸. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||||||||||||||||||||||||
° v |
|
|
5v |
2 |
2v |
|
|
|
0; |
|
|
|
|
° |
|
v z 0; |
|
|
|
¨ |
|
2 |
¸ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||||||||
¯ 1 |
|
|
|
|
|
|
3 |
|
|
|
|
|
|
|
|
¯ |
|
1 |
|
|
|
|
© |
|
|
|
|
|
¹ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||
Ⱥɧɚɥɨɝɢɱɧɨ ɧɚɯɨɞɢɦ ɫɨɛɫɬɜɟɧɧɵɟ ɜɟɤɬɨɪɵ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||||||||||||||||||||||||||||
|
|
|
§ |
1 · |
|
|
|
|
|
|
|
§ |
0 |
· |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||
V2 |
|
¨ |
|
|
¸ |
|
V3 |
|
|
¨ |
1 |
¸ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||
|
¨ |
1¸, |
|
|
¨ |
¸, |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||||||||
|
|
|
¨ |
|
|
¸ |
|
|
|
|
|
|
|
¨ |
1 |
¸ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
|
|
|
© |
1¹ |
|
|
|
|
|
|
|
© |
¹ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||
ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ k2 |
|
|
1, |
|
k3 |
|
2 . |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||||||||||||
Ɉɛɳɟɟ ɪɟɲɟɧɢɟ ɫɢɫɬɟɦɵ ɬɚɤɨɜɨ: |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
§ |
1 · |
|
|
|
|
|
|
§ |
1 · |
|
|
|
|
§ |
0· |
|
|
|
|
|||||
|
|
|
|
|
|
|
|
k t |
C V |
|
k |
t |
|
|
|
k t |
C |
¨ |
1 |
¸ |
e |
t |
C |
|
¨ |
1 |
¸ |
e |
t |
C |
|
¨ |
1 |
¸ |
e |
2t |
; |
|||||||||||||
|
X C V e 1 |
e 2 |
|
|
C V e 3 |
|
|
¸ |
|
|
2 |
¨ |
¸ |
|
3 |
¨ |
¸ |
|
||||||||||||||||||||||||||||||||
|
|
|
|
1 1 |
|
|
|
|
|
|
|
2 2 |
|
|
|
|
|
|
3 3 |
1 ¨ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
¨ |
|
¸ |
|
|
|
|
|
|
¨ |
|
¸ |
|
|
|
|
¨ |
1 |
¸ |
|
|
|
|
||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
© 2 |
¹ |
|
|
|
|
|
|
© |
1¹ |
|
|
|
|
© |
¹ |
|
|
|
|
|||||
ɢɥɢ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
x |
C e t C |
2 |
et ; |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||||||||
|
|
|
1 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
y C e t C |
2 |
et C |
|
e2t ; |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||||||||||||||
|
|
|
|
1 |
|
|
|
|
|
|
|
|
|
|
|
|
3 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
|
z 2C e t |
C |
2 |
et |
C |
e2t . |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||||||||||||||
|
|
|
|
|
1 |
|
|
|
|
|
|
|
|
|
|
|
3 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||
Ⱦɥɹ |
ɧɚɯɨɠɞɟɧɢɹ |
ɱɚɫɬɧɨɝɨ |
ɪɟɲɟɧɢɹ |
|
ɩɨɞɫɬɚɜɢɦ |
ɜ ɨɛɳɟɟ |
ɪɟɲɟɧɢɟ t 0 , |
|||||||||||||||||||||||||||||||||||||||||||
x 1, y |
2, z |
|
|
0ɢ ɨɩɪɟɞɟɥɢɦ C1, C2 , C3 ɢɡ ɩɨɥɭɱɟɧɧɨɣ ɫɢɫɬɟɦɵ: |
|
|
|
|
|
|
|
|
||||||||||||||||||||||||||||||||||||||
76
1 |
C |
C |
; |
C |
2; |
° |
1 |
2 |
|
° 1 |
3; |
® 2 C1 C2 C3 |
; ®ɋ2 |
||||
° |
2C1 C2 C3 |
° |
1. |
||
¯C |
¯C3 |
||||
ɂɫɤɨɦɨɟ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ
x 2e t 3et ; y 2e t 3et e2t ; z 4e t 3et e2t .
|
|
|
|
|
|
|
|
dx |
2x 3y; |
|
|
|
||||
|
|
|
|
|
|
|
|
° |
|
|
|
|
|
|||
|
|
|
|
|
|
|
|
|
|
|
|
|
||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
ɉɪɢɦɟɪ 7.4. ɇɚɣɬɢ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɫɢɫɬɟɦɵ ®dt |
|
|
|
|
|
|
||||||||||
|
|
|
|
|
|
|
|
° |
dy |
3x 2 y. |
|
|
|
|||
|
|
|
|
|
|
|
|
|
|
|
|
|
||||
|
|
|
|
|
|
|
|
¯dt |
|
|
|
|
|
|
||
Ɋɟɲɟɧɢɟ. ɏɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ |
|
|
|
|
|
|
|
|
|
|||||||
|
2 k |
3 |
|
0; k 2 |
4k 13 |
0 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||
|
3 |
2 k |
|
|
|
|
|
|
|
|
|
|
||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
ɢɦɟɟɬ ɤɨɪɧɢ k 2 3i, k |
|
2 3i . |
ɇɚɯɨɞɢɦ |
ɫɨɛɫɬɜɟɧɧɵɣ |
ɜɟɤɬɨɪ V |
§v |
· |
|||||||||
2 |
¨ 1 |
¸, |
||||||||||||||
|
|
1 |
|
|
|
|
|
|
|
|
|
|
1 |
¨ |
¸ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
©v2 |
¹ |
ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɣ ɤɨɪɧɸ k1 2 3i |
ɢɡ ɫɢɫɬɟɦɵ: |
3iv1 3v2 |
0, |
ɉɨɥɚɝɚɹ v1 |
1 , |
|||||||||||
® |
3v |
3iv |
|
0. |
||||||||||||
|
|
|
|
|
|
|
|
¯ |
1 |
|
2 |
|
|
|
|
|
ɩɨɥɭɱɢɦ v |
|
i V |
§ |
1 · |
|
|
|
|
||
|
¨ |
¸. ɋɨɫɬɚɜɢɦ ɜɵɪɚɠɟɧɢɟ |
|
|||||||
|
2 |
|
|
1 |
© |
i ¹ |
|
|
|
|
k t |
§ |
1 · |
(2 3i)t |
§ 1 |
· |
2t |
(cos3t i sin 3t) |
§e |
||
V e 1 |
|
¨ |
¸e |
|
|
¨ |
¸e |
|
¨ |
|
1 |
|
© i ¹ |
|
|
© i ¹ |
|
|
¨ |
||
|
|
|
|
|
|
©e |
||||
2t
2t
(cos3t i sin 3t)·¸ (sin 3t i cos3t)¸¹.
Ɂɞɟɫɶ |
ɢɫɩɨɥɶɡɨɜɚɧɚ |
|
ɮɨɪɦɭɥɚ |
e(D iE)t |
eDt (cos Et isin Et) . ɋɨɝɥɚɫɧɨ |
|||||||
ɡɚɦɟɱɚɧɢɸ, ɞɜɚ ɱɚɫɬɧɵɯ ɪɟɲɟɧɢɹ ɢɫɯɨɞɧɨɣ ɫɢɫɬɟɦɵ ɢɦɟɸɬ ɜɢɞ |
||||||||||||
Re(V ek1t ) |
§ |
2t |
cos3t |
· |
Im(V ek1t ) |
§ |
e |
2t |
|
· |
||
¨e |
2t |
¸, |
¨ |
|
|
cos3t ¸. |
||||||
1 |
|
¨ |
sin 3t |
¸ |
1 |
¨ |
e |
2t |
¸ |
|||
|
|
©e |
|
¹ |
|
© |
|
|
sin 3t ¹ |
|||
Ɉɛɳɢɦ ɪɟɲɟɧɢɟɦ ɫɢɫɬɟɦɵ ɛɭɞɟɬ
77
X |
§ x |
· |
|
C |
Re(V ek1t ) C |
|
Im(V ek1t ) |
C |
§e2t |
cos3t · |
C |
|
§ |
e2t cos3t |
· |
||||||
¨ |
¸ |
|
2 |
¨ |
2t |
¸ |
2 |
¨ |
|
2t |
|
¸ |
|||||||||
|
© y ¹ |
|
1 |
1 |
|
|
|
1 |
1 |
¨ |
¸ |
|
¨ |
e |
|
¸ |
|||||
|
|
|
|
|
|
|
|
|
|
©e |
|
sin 3t ¹ |
|
|
© |
|
sin 3t ¹ |
||||
ɢɥɢ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
2t |
cos3t C2e |
2t |
sin 3t; |
|
|
|
|
|
|
|
|
|
|
|
||||
°x C1e |
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||||
® |
|
|
2t |
|
|
2t |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
° |
C1e |
sin 3t C2e |
cos3t. |
|
|
|
|
|
|
|
|
|
|
|
|||||||
¯y |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||||
7.3. Ɂɚɞɚɱɢ ɞɢɧɚɦɢɤɢ, ɩɪɢɜɨɞɹɳɢɟ ɤ ɪɟɲɟɧɢɸ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ
Ʉ ɡɚɞɚɱɟ ɞɢɧɚɦɢɤɢ ɬɨɱɤɢ, ɩɪɢɜɨɞɹɳɟɣ ɤ ɪɟɲɟɧɢɸ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ, ɨɬɧɨɫɹɬɫɹ ɬɟ ɡɚɞɚɱɢ, ɜ ɤɨɬɨɪɵɯ ɨɩɪɟɞɟɥɹɟɬɫɹ ɞɜɢɠɟɧɢɟ ɬɨɱɤɢ ɩɨ ɡɚɞɚɧɧɵɦ ɫɢɥɚɦ. ɋɢɥɵ, ɞɟɣɫɬɜɭɸɳɢɟ ɧɚ ɬɨɱɤɭ, ɦɨɝɭɬ ɛɵɬɶ ɤɚɤ ɩɨɫɬɨɹɧɧɵɦɢ, ɬɚɤ ɢ ɡɚɞɚɧɧɵɦɢ ɮɭɧɤɰɢɹɦɢ ɜɪɟɦɟɧɢ, ɤɨɨɪɞɢɧɚɬ, ɫɤɨɪɨɫɬɢ, ɬɨ ɟɫɬɶ
Fx Fx (t, x, y, z, x, y, z); Fy Fy (t, x, y, z, x, y, z); Fz Fz (t, x, y, z, x, y, z).
Ɋɟɲɟɧɢɟ ɬɚɤɢɯ ɡɚɞɚɱ ɫɜɨɞɢɬɫɹ ɤ ɢɧɬɟɝɪɢɪɨɜɚɧɢɸ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɞɜɢɠɟɧɢɹ ɬɨɱɤɢ: ɜ ɤɨɨɪɞɢɧɚɬɧɨɣ ɮɨɪɦɟ
|
|
Fx ; |
° |
mx |
|
|
Fy ; |
|
®my |
||
°mz |
F , |
|
¯ |
|
z |
ɢɥɢ ɜ ɟɫɬɟɫɬɜɟɧɧɨɣ ɮɨɪɦɟ
m dv |
F ; |
|||
° |
dt |
t |
||
° |
v2 |
|
||
° |
|
|||
®m |
|
|
Fh ; |
|
U |
||||
° |
|
|||
°O |
F . |
|
||
° |
|
b |
|
|
¯ |
|
|
|
|
ɫɢɫɬɟɦɵ
(7.1)
(7.2)
78
ȼ ɷɬɢɯ ɭɪɚɜɧɟɧɢɹɯ ɩɨɞ F ɩɨɧɢɦɚɟɬɫɹ ɪɚɜɧɨɞɟɣɫɬɜɭɸɳɚɹ ɜɫɟɯ ɫɢɥ, ɜ ɬɨɦ ɱɢɫɥɟ ɢ ɪɟɚɤɰɢɣ ɫɜɹɡɟɣ, ɟɫɥɢ ɬɨɱɤɚ ɧɟ ɫɜɨɛɨɞɧɚ. ɉɪɢ ɢɧɬɟɝɪɢɪɨɜɚɧɢɢ ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɣ (7.1) ɜ ɨɛɳɟɦ ɫɥɭɱɚɟ ɩɨɹɜɥɹɟɬɫɹ ɲɟɫɬɶ ɩɪɨɢɡɜɨɥɶɧɵɯ ɩɨɫɬɨɹɧɧɵɯ, ɤɨɬɨɪɵɟ ɨɩɪɟɞɟɥɹɸɬɫɹ ɩɨ ɧɚɱɚɥɶɧɵɦ ɭɫɥɨɜɢɹɦ. ɉɨɞ ɧɚɱɚɥɶɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɞɜɢɠɟɧɢɹ ɬɨɱɤɢ ɩɨɧɢɦɚɸɬɫɹ ɡɧɚɱɟɧɢɹ ɤɨɨɪɞɢɧɚɬ ɢ ɩɪɨɟɤɰɢɣ ɫɤɨɪɨɫɬɢ ɬɨɱɤɢ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɞɜɢɠɟɧɢɹ, ɬɨ ɟɫɬɶ ɩɪɢ t 0
x x0 ; vx x0 ; y y0 ; vy y0 ; z z0 ; vz z0.
ȿɫɥɢ ɞɜɢɠɟɧɢɟ ɬɨɱɤɢ ɩɪɨɢɫɯɨɞɢɬ ɧɚ ɩɥɨɫɤɨɫɬɢ, ɬɨ ɱɢɫɥɨ ɭɪɚɜɧɟɧɢɣ (7.1) ɫɨɤɪɚɳɚɟɬɫɹ ɞɨ ɞɜɭɯ, ɚ ɱɢɫɥɨ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɣ ɞɨ ɱɟɬɵɪɟɯ. ɉɪɢ ɞɜɢɠɟɧɢɢ ɬɨɱɤɢ ɩɨ ɩɪɹɦɨɣ ɛɭɞɟɦ ɢɦɟɬɶ ɨɞɧɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɢ ɞɜɚ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɹ.
ɉɪɢ ɪɟɲɟɧɢɢ ɡɚɞɚɱ ɩɨɥɟɡɧɨ ɩɪɢɞɟɪɠɢɜɚɬɶɫɹ ɫɥɟɞɭɸɳɟɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ.
1. ɋɨɫɬɚɜɢɬɶ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ.
ɚ) ɜɵɛɪɚɬɶ ɤɨɨɪɞɢɧɚɬɧɵɟ ɨɫɢ, ɩɨɦɟɫɬɢɜ ɢɯ ɧɚɱɚɥɨ ɜ ɧɚɱɚɥɶɧɨɟ ɩɨɥɨɠɟɧɢɟ ɬɨɱɤɢ; ɟɫɥɢ ɞɜɢɠɟɧɢɟ ɬɨɱɤɢ ɹɜɥɹɟɬɫɹ ɩɪɹɦɨɥɢɧɟɣɧɵɦ, ɬɨ ɨɞɧɭ ɢɡ ɤɨɨɪɞɢɧɚɬɧɵɯ ɨɫɟɣ ɫɥɟɞɭɟɬ ɩɪɨɜɨɞɢɬɶ ɜɞɨɥɶ ɥɢɧɢɢ ɞɜɢɠɟɧɢɹ ɬɨɱɤɢ; ɛ) ɢɡɨɛɪɚɡɢɬɶ ɞɜɢɠɭɳɭɸɫɹ ɬɨɱɤɭ ɜ ɩɪɨɢɡɜɨɥɶɧɵɣ ɬɟɤɭɳɢɣ ɦɨɦɟɧɬ t ɢ ɩɨɤɚɡɚɬɶ ɧɚ ɪɢɫɭɧɤɟ ɜɫɟ ɞɟɣɫɬɜɭɸɳɢɟ ɧɚ ɧɟɟ ɫɢɥɵ, ɜ ɬɨɦ ɱɢɫɥɟ ɢ ɪɟɚɤɰɢɢ ɫɜɹɡɟɣ, ɩɪɢ ɧɚɥɢɱɢɢ ɫɢɥ, ɡɚɜɢɫɹɳɢɯ ɨɬ ɫɤɨɪɨɫɬɢ, ɜɟɤɬɨɪ ɫɤɨɪɨɫɬɢ ɧɚɩɪɚɜɢɬɶ ɩɪɟɞɩɨɥɨɠɢɬɟɥɶɧɨ ɬɚɤ, ɱɬɨɛɵ ɜɫɟ ɟɝɨ ɩɪɨɟɤɰɢɢ ɧɚ ɜɵɛɪɚɧɧɵɟ ɨɫɢ ɛɵɥɢ ɩɨɥɨɠɢɬɟɥɶɧɵɦɢ; ɜ) ɧɚɣɬɢ ɫɭɦɦɭ ɩɪɨɟɤɰɢɣ ɜɫɟɯ ɫɢɥ ɧɚ ɜɵɛɪɚɧɧɵɟ ɨɫɢ ɢ ɩɨɞɫɬɚɜɢɬɶ ɷɬɭ ɫɭɦɦɭ ɜ ɩɪɚɜɵɟ ɱɚɫɬɢ ɭɪɚɜɧɟɧɢɣ
(7.1).
79
2.ɉɪɨɢɧɬɟɝɪɢɪɨɜɚɬɶ ɩɨɥɭɱɟɧɧɵɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ. ɂɧɬɟɝɪɢɪɨɜɚɧɢɟ ɩɪɨɢɡɜɨɞɢɬɫɹ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɦɢ ɦɟɬɨɞɚɦɢ, ɡɚɜɢɫɹɳɢɦɢ ɨɬ ɜɢɞɚ ɩɨɥɭɱɟɧɧɵɯ ɭɪɚɜɧɟɧɢɣ.
3.ɍɫɬɚɧɨɜɢɬɶ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ ɞɜɢɠɟɧɢɹ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɢ ɩɨ ɧɢɦ ɨɩɪɟɞɟɥɢɬɶ ɩɪɨɢɡɜɨɥɶɧɵɟ ɩɨɫɬɨɹɧɧɵɟ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ.
4.ɂɡ ɩɨɥɭɱɟɧɧɵɯ ɜ ɪɟɡɭɥɶɬɚɬɟ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɭɪɚɜɧɟɧɢɣ ɨɩɪɟɞɟɥɢɬɶ ɢɫɤɨɦɵɟ ɜɟɥɢɱɢɧɵ.
Ɂɚɦɟɱɚɧɢɟ 1. ɉɪɢ ɢɧɬɟɝɪɢɪɨɜɚɧɢɢ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɢɧɨɝɞɚ ɰɟɥɟɫɨɨɛɪɚɡɧɨ ɨɩɪɟɞɟɥɢɬɶ ɡɧɚɱɟɧɢɹ ɩɪɨɢɡɜɨɥɶɧɵɯ ɩɨɫɬɨɹɧɧɵɯ ɩɨ ɦɟɪɟ ɢɯ ɩɨɹɜɥɟɧɢɹ.
ɉɪɢɦɟɪ 7.5. Ⱥɜɬɨɦɨɛɢɥɶ, ɦɚɫɫɵ m ɞɜɢɠɟɬɫɹ ɩɪɹɦɨɥɢɧɟɣɧɨ ɢɡ ɫɨɫɬɨɹɧɢɹ ɩɨɤɨɹ ɢ ɢɦɟɟɬ ɞɜɢɝɚɬɟɥɶ, ɤɨɬɨɪɵɣ ɪɚɡɜɢɜɚɟɬ ɩɨɫɬɨɹɧɧɭɸ ɬɹɝɭ F, ɧɚɩɪɚɜɥɟɧɧɭɸ ɜ ɫɬɨɪɨɧɭ ɞɜɢɠɟɧɢɹ, ɞɨ ɩɨɥɧɨɝɨ ɫɝɨɪɚɧɢɹ ɝɨɪɸɱɟɝɨ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ Ɍ, ɩɨɫɥɟ ɱɟɝɨ ɚɜɬɨɦɨɛɢɥɶ ɞɜɢɠɟɬɫɹ ɩɨ ɢɧɟɪɰɢɢ ɞɨ ɨɫɬɚɧɨɜɤɢ. ɇɚɣɬɢ ɩɪɨɣɞɟɧɧɵɣ ɩɭɬɶ. ɋɢɥɭ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɫɱɢɬɚɬɶ ɩɨɫɬɨɹɧɧɨɣ ɢ ɪɚɜɧɨɣ R. ɂɡɦɟɧɟɧɢɟɦ ɦɚɫɫɵ ɚɜɬɨɦɨɛɢɥɹ ɩɪɟɧɟɛɪɟɱɶ.
|
|
Ɋɟɲɟɧɢɟ. ȼɟɫɶ ɩɭɬɶ S ɫɤɥɚɞɵɜɚɟɬɫɹ ɢɡ S1 | |
AC |, ɧɚ ɤɨɬɨɪɨɦ ɞɟɣɫɬɜɭɟɬ ɫɢɥɚ |
||||||||||||||
F ɞɨ ɩɨɥɧɨɝɨ ɫɝɨɪɚɧɢɹ ɝɨɪɸɱɟɝɨ ɢ S2 |
| CB |, |
ɤɨɬɨɪɵɣ |
ɚɜɬɨɦɨɛɢɥɶ ɢɞɟɬ |
ɩɨ |
|||||||||||||
ɢɧɟɪɰɢɢ. ɇɚ ɩɭɬɢ Ⱥɋ: |
mx |
F R ; |
|
|
|
|
(7.3) |
|
|||||||||
ɧɚ ɩɭɬɢ |
ɋȼ: |
mx |
|
|
|
|
|
|
|
|
|
(7.4) |
|
||||
|
R . |
|
|
|
|
|
|
|
|||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
³mdx |
³(F R)dt ; |
|
|
|
|
Ɋɟɲɢɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ (7.3): |
|
||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
mx |
|
(F R)t C1 ; ɩɪɢ t |
0 ɛɭɞɟɬ x |
0 , ɨɬɤɭɞɚ |
|
|
|
||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
C1 |
|
0 mx |
(F R)t . |
|
|
|
|
|
|
(7.5) |
|
||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
(F R)t2 |
|
|
|
|
||
|
|
ɂɧɬɟɝɪɢɪɭɹ, |
ɩɨɥɭɱɢɦ |
mx |
|
|
C2 ; |
ɩɪɢ |
t |
0 ɛɭɞɟɬ x 0 , ɨɬɤɭɞɚ |
|||||||
|
|
2 |
|
||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
0 ; |
x |
|
(F R)t2 |
|
|
|
|
|
|
|
|
|
|||
C |
2 |
|
|
|
|
. |
Ɉɩɪɟɞɟɥɢɦ ɩɭɬɶ |
S , ɤɨɬɨɪɵɣ |
ɩɪɨɣɞɟɬ ɚɜɬɨɦɨɛɢɥɶ |
ɞɨ |
|||||||
|
|
|
|
||||||||||||||
|
|
|
|
|
2m |
|
|
|
|
|
1 |
|
|
|
|
||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
80